A hollow sphere has radius 6.4 m. Minimum velocity required by a motor cyclist at bottom to complete the circle will be.

Options

A. `17.7 m//s`
B. `10.2 m//s`
C. `12.4 m//s`
D. `16.0 m//s`

Explanation

To find the minimum velocity required by a motorcyclist at the bottom of a hollow sphere with a radius of 6.4 m to complete the circular path, we can follow these steps:

### Step 1: Understand the Forces at the Top of the Sphere
At the top of the sphere, the motorcyclist must have enough velocity to ensure that the gravitational force provides the necessary centripetal force to keep moving in a circle. The centripetal force required is given by:

\[ F_c = \frac{mv^2}{R} \]

Where:
- \( F_c \) is the centripetal force,
- \( m \) is the mass of the motorcyclist,
- \( v \) is the velocity at the top,
- \( R \) is the radius of the sphere.

At the top of the sphere, the only forces acting on the motorcyclist are the gravitational force and the centripetal force. Thus, we can set up the equation:

\[ mg = \frac{mv^2}{R} \]

### Step 2: Simplify the Equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):

\[ g = \frac{v^2}{R} \]

Rearranging gives us:

\[ v^2 = gR \]

### Step 3: Substitute the Known Values
We know that \( g \) (acceleration due to gravity) is approximately \( 10 \, \text{m/s}^2 \) and the radius \( R \) is \( 6.4 \, \text{m} \). Substituting these values into the equation:

\[ v^2 = 10 \times 6.4 \]

Calculating this gives:

\[ v^2 = 64 \implies v = \sqrt{64} = 8 \, \text{m/s} \]

### Step 4: Use Conservation of Energy
To find the minimum velocity at the bottom of the sphere, we can use the conservation of energy principle. The total mechanical energy at the bottom must equal the total mechanical energy at the top.

At the bottom:
- Kinetic Energy (KE) = \( \frac{1}{2} mv^2 \)
- Potential Energy (PE) = 0 (taking the bottom as the reference point)

At the top:
- Kinetic Energy (KE) = \( \frac{1}{2} mv_0^2 \)
- Potential Energy (PE) = \( mg(2R) \)

Setting the total energy at the bottom equal to that at the top:

\[ \frac{1}{2} mv^2 = \frac{1}{2} mv_0^2 + mg(2R) \]

### Step 5: Substitute and Rearrange
Substituting \( v_0^2 = gR \) into the equation gives:

\[ \frac{1}{2} mv^2 = \frac{1}{2} m(gR) + mg(2R) \]

Cancelling \( m \) from both sides:

\[ \frac{1}{2} v^2 = \frac{1}{2} gR + 2gR \]

This simplifies to:

\[ \frac{1}{2} v^2 = \frac{1}{2} gR + 4gR = \frac{9}{2} gR \]

Multiplying both sides by 2:

\[ v^2 = 9gR \]

### Step 6: Substitute the Values Again
Now substituting \( g = 10 \, \text{m/s}^2 \) and \( R = 6.4 \, \text{m} \):

\[ v^2 = 9 \times 10 \times 6.4 = 576 \]

Taking the square root gives:

\[ v = \sqrt{576} = 24 \, \text{m/s} \]

### Conclusion
Thus, the minimum velocity required by the motorcyclist at the bottom of the hollow sphere to complete the circular path is:

\[ \boxed{24 \, \text{m/s}} \]